Mathematics • Year 9 • Unit 4 • Lesson 16
Probability in the Real World
Use experimental and theoretical probability in real Y9 contexts: insurance, weather, quality control, sport and gambling. Then explain in full sentences how the law of large numbers protects casinos.
1. Word problems
Each scenario uses one of the lesson's two formulas: theoretical P = favourable ÷ total, or experimental P ≈ occurrences ÷ trials. Show every step.
1.1 — Insurance premiums. An insurance company looks at 50 000 home policies last year. 1 250 households made a claim.
(a) Find the experimental probability that a household makes a claim in a year.
(b) Out of 8 000 new policies this year, how many claims would the company expect to pay?
(c) Why does the company use experimental rather than theoretical probability here? 4 marks
1.2 — Quality control on chocolate bars. A factory inspector samples 400 chocolate bars and finds 6 are underweight.
(a) Find the experimental P(underweight) as a decimal.
(b) If the factory makes 25 000 bars per shift, how many underweight bars would you expect per shift?
(c) The factory only fails inspection if more than 1% of bars are underweight. Did this sample fail? Show how you decided. 4 marks
1.3 — Weather forecasting. Over the last 20 Junes in Sydney, it rained on 1 June on exactly 6 of those years.
(a) Find the experimental P(rain on 1 June in Sydney).
(b) A weather presenter says "there's a 50% chance of rain tomorrow". Explain in one sentence why this is closer to experimental than theoretical probability.
(c) Out of the next 100 Junes, roughly how many would you expect to have rain on 1 June, based on this data? 3 marks
1.4 — Basketball free throws. A Year 9 player takes 80 free throws in training. She scores 56.
(a) Find her experimental P(score).
(b) In a game she gets 25 free throws. How many would she expect to score?
(c) She only actually scores 14 of the 25 in the game. Find the new experimental P(score) for that game, then explain in one sentence why it differs from her training value. 4 marks
1.5 — Pokie machine. A poker machine is set up so that the theoretical probability of any single spin paying out the top prize is 1 in 50 000.
(a) How many top-prize wins would the venue expect if 600 000 spins are played in a year?
(b) The venue actually pays out the top prize 9 times that year. Find the experimental probability of a top-prize win and compare it with the theoretical value.
(c) Does this evidence suggest the machine is rigged? Justify in one sentence. 4 marks
2. Explain your thinking
This is a communication task. Use full sentences. 4 marks
2.1 Your friend says "If I keep playing the pokies long enough I'll definitely come out ahead, because the law of large numbers means I'll start winning eventually." Explain, using ideas from Lesson 16, why this is wrong. In your answer:
(i) State what the law of large numbers actually says (one sentence).
(ii) Explain who the law of large numbers actually favours: the player or the venue, and why.
(iii) Use the words "theoretical", "experimental" and "house edge" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Insurance premiums
(a) P(claim) = 1 250 ÷ 50 000 = 0.025 (2.5%).
(b) Expected claims = 8 000 × 0.025 = 200 claims.
(c) Insurance uses experimental probability because there is no clean theoretical model for "will this household have a fire/theft/storm" — only past data on what actually happens. Decades of real claims data is far more reliable than guessing equally likely outcomes.
1.2 — Quality control
(a) P(underweight) = 6 ÷ 400 = 0.015.
(b) Expected per shift = 25 000 × 0.015 = 375 underweight bars.
(c) Yes — 0.015 = 1.5%, which is above the 1% threshold. The factory fails this inspection.
1.3 — Weather forecasting
(a) Experimental P(rain on 1 June) = 6 ÷ 20 = 0.3 (30%).
(b) Weather doesn't have equally likely outcomes you can list — forecasts come from historical patterns and atmospheric data, so 50% is experimental in nature, not theoretical.
(c) Expect about 0.3 × 100 = 30 of the next 100 Junes to have rain on 1 June (assuming long-run conditions are similar).
1.4 — Basketball free throws
(a) Training P(score) = 56 ÷ 80 = 0.7 (70%).
(b) Expected = 25 × 0.7 = 17.5, so about 17–18 scores.
(c) Game P(score) = 14 ÷ 25 = 0.56. Lower than training because of game pressure, defenders, fatigue, and because 25 throws is a small sample — the law of large numbers needs more trials to wash out random variation.
1.5 — Pokie machine
(a) Expected wins = 600 000 ÷ 50 000 = 12 top-prize wins.
(b) Experimental P = 9 ÷ 600 000 = 1.5 × 10⁻⁵ = 1 in ~66 667. Compared with theoretical 1 in 50 000, the machine has paid out slightly less than expected.
(c) No — 9 wins vs 12 expected is a small difference that random variation easily explains. There is no strong evidence of rigging from this data alone.
2.1 — Explain your thinking (sample response)
The law of large numbers says that as the number of trials increases, experimental probability gets closer to the theoretical probability — it does NOT say players will eventually start winning. On a pokie, the theoretical probability of each game is set so that the venue keeps a small percentage of every dollar wagered — this is the house edge. The more times you play, the closer the experimental return-to-player drops to this theoretical losing rate. So the law of large numbers actually favours the venue, not the player: the longer you play, the more certain it becomes that you lose. A short lucky streak can happen with random variation, but with many trials it gets averaged away by the consistent house edge.
Marking: 1 mark for correctly stating the law of large numbers; 1 for correctly identifying that it favours the house; 1 for correct use of "theoretical / experimental"; 1 for "house edge" or equivalent explanation.